Problem: $ C = \left[\begin{array}{rr}1 & 0 \\ -1 & 3\end{array}\right]$ $ A = \left[\begin{array}{rrr}4 & 1 & -2 \\ -1 & 0 & 1\end{array}\right]$ What is $ C A$ ?
Because $ C$ has dimensions $(2\times2)$ and $ A$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ C A = \left[\begin{array}{rr}{1} & {0} \\ {-1} & {3}\end{array}\right] \left[\begin{array}{rrr}{4} & \color{#DF0030}{1} & \color{#9D38BD}{-2} \\ {-1} & \color{#DF0030}{0} & \color{#9D38BD}{1}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ A$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ A$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ A$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{1}\cdot{4}+{0}\cdot{-1} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ A$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{4}+{0}\cdot{-1} & ? & ? \\ {-1}\cdot{4}+{3}\cdot{-1} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ A$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{4}+{0}\cdot{-1} & {1}\cdot\color{#DF0030}{1}+{0}\cdot\color{#DF0030}{0} & ? \\ {-1}\cdot{4}+{3}\cdot{-1} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{1}\cdot{4}+{0}\cdot{-1} & {1}\cdot\color{#DF0030}{1}+{0}\cdot\color{#DF0030}{0} & {1}\cdot\color{#9D38BD}{-2}+{0}\cdot\color{#9D38BD}{1} \\ {-1}\cdot{4}+{3}\cdot{-1} & {-1}\cdot\color{#DF0030}{1}+{3}\cdot\color{#DF0030}{0} & {-1}\cdot\color{#9D38BD}{-2}+{3}\cdot\color{#9D38BD}{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}4 & 1 & -2 \\ -7 & -1 & 5\end{array}\right] $